11.1 Laplace Transforms of $U[x]$, $U'[x]$, and $U”[x]$

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The following three plots illustrate the Laplace transforms of the distributional representations of $U(x)$, $U'(x)$, and $U”(x)$ defined in formulas (4), (5), and (6) on page 10 above in blue and the Laplace transforms of the Fourier series representations of $U(x)$, $U'(x)$, and $U”(x)$ in orange. The three plots below correspond to evaluation of the Laplace transforms of $U(x)$, $U'(x)$ and $U”(x)$ from $y=0$ to $y=3$.

$\mathcal{L}_x[U(x)](y)=\frac{e^{-y}}{y}$ from $y=0$ to $y=3$
$\mathcal{L}_x[U'(x)](y)=e^{-y}$ from $y=0$ to $y=3$
$\mathcal{L}_x[U”(x)](y)=y\,e^{-y}$ from $y=0$ to $y=3$


The following two plots illustrate the Laplace transform of the distributional representation of $U(x)$ defined in formula (4) on page 10 above in blue and the Laplace transform of the Fourier series representation of $U(x)$ in orange. The two plots below correspond to evaluation of the real and imaginary parts of the Laplace transform of $U(x)$ evaluated along the line $y=\frac{1}{2}+i\, t$.

$\Re[\mathcal{L}_x[U(x)](y)]$ evaluated along the line $y=\frac{1}{2}+i\,t$
$\Im[\mathcal{L}_x[U(x)](y)]$ evaluated along the line $y=\frac{1}{2}+i\,t$


The following two plots illustrate the Laplace transform of the distributional representation of $U'(x)$ defined in formula (5) on page 10 above in blue and the Laplace transform of the Fourier series representation of $U'(x)$ in orange. The two plots below correspond to evaluation of the real and imaginary parts of the Laplace transform of $U'(x)$ evaluated along the line $y=\frac{1}{2}+i\, t$.

$\Re[\mathcal{L}_x[U'(x)](y)]$ evaluated along the line $y=\frac{1}{2}+i\,t$
$\Im[\mathcal{L}_x[U'(x)](y)]$ evaluated along the line $y=\frac{1}{2}+i\,t$


The following two plots illustrate the Laplace transform of the distributional representation of $U”(x)$ defined in formula (6) on page 10 above in blue and the Laplace transform of the Fourier series representation of $U”(x)$ in orange. The two plots below correspond to evaluation of the real and imaginary parts of the Laplace transform of $U”(x)$ evaluated along the line $y=\frac{1}{2}+i\, t$.

$\Re[\mathcal{L}_x[U”(x)](y)]$ evaluated along the line $y=\frac{1}{2}+i\,t$
$\Im[\mathcal{L}_x[U”(x)](y)]$ evaluated along the line $y=\frac{1}{2}+i\,t$